A helpful tip for integration by parts

Thread Starter


Joined Nov 13, 2010
There comes a time and a place when a difficult integration can become an easy one, i.e.

\( F(x) = \int u\ dv = uv - \int v\;du \)


\( F(x) = \int\;x^9cos\;ax\;dx\)

You will need to perform integration by parts TEN times in order to reduce the variable "x" and its exponent to a constant. This is (by hand) horrendous computation.

Let's integrate,

\( G(x) = \int\; x^3\;cos\;ax\;dx\)


\( G(x) = \frac{x^3\;sin\;ax}{a} + \frac{(3x^2)\;cos\;ax}{a^2} - \frac{(6x)\;sin\;ax}{a^3} - \frac{(6)cos\;ax}{a^4} + C\)

Can you see the pattern?

The first term is uv, for each term there after, u is differentiated, v is integrated and the result is multiplied by minus 1 for obvious reasons in the integration by parts formula.

Given any product of the form,

\( F(x) = \int\;(a_nx^n + a_{n-1} x^{n-1}+... + a_{0}x^{0})cos\;ax\;dx \)

Where v can be a sine or cosine, can be integrated in less than one minute, provided the exponent is within reason of course.

Let's return to the first example

\( F(x) = \int\;x^9cos\;ax\;dx\)

and now use the method claimed.

You will be able to integrate this (by hand) in less than one minute I assure you.
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